Runtime Complexity TRS:
The TRS R consists of the following rules:

from(X) → cons(X, n__from(s(X)))
first(0, Z) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
sel(0, cons(X, Z)) → X
sel(s(X), cons(Y, Z)) → sel(X, activate(Z))
from(X) → n__from(X)
first(X1, X2) → n__first(X1, X2)
activate(n__from(X)) → from(X)
activate(n__first(X1, X2)) → first(X1, X2)
activate(X) → X

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

Runtime Complexity TRS:
The TRS R consists of the following rules:

from'(X) → cons'(X, n__from'(s'(X)))
first'(0', Z) → nil'
first'(s'(X), cons'(Y, Z)) → cons'(Y, n__first'(X, activate'(Z)))
sel'(0', cons'(X, Z)) → X
sel'(s'(X), cons'(Y, Z)) → sel'(X, activate'(Z))
from'(X) → n__from'(X)
first'(X1, X2) → n__first'(X1, X2)
activate'(n__from'(X)) → from'(X)
activate'(n__first'(X1, X2)) → first'(X1, X2)
activate'(X) → X

Rewrite Strategy: INNERMOST

Infered types.

Rules:
from'(X) → cons'(X, n__from'(s'(X)))
first'(0', Z) → nil'
first'(s'(X), cons'(Y, Z)) → cons'(Y, n__first'(X, activate'(Z)))
sel'(0', cons'(X, Z)) → X
sel'(s'(X), cons'(Y, Z)) → sel'(X, activate'(Z))
from'(X) → n__from'(X)
first'(X1, X2) → n__first'(X1, X2)
activate'(n__from'(X)) → from'(X)
activate'(n__first'(X1, X2)) → first'(X1, X2)
activate'(X) → X

Types:
from' :: s':0' → n__from':cons':nil':n__first'
cons' :: s':0' → n__from':cons':nil':n__first' → n__from':cons':nil':n__first'
n__from' :: s':0' → n__from':cons':nil':n__first'
s' :: s':0' → s':0'
first' :: s':0' → n__from':cons':nil':n__first' → n__from':cons':nil':n__first'
0' :: s':0'
nil' :: n__from':cons':nil':n__first'
n__first' :: s':0' → n__from':cons':nil':n__first' → n__from':cons':nil':n__first'
activate' :: n__from':cons':nil':n__first' → n__from':cons':nil':n__first'
sel' :: s':0' → n__from':cons':nil':n__first' → s':0'
_hole_n__from':cons':nil':n__first'1 :: n__from':cons':nil':n__first'
_hole_s':0'2 :: s':0'
_gen_n__from':cons':nil':n__first'3 :: Nat → n__from':cons':nil':n__first'
_gen_s':0'4 :: Nat → s':0'

Heuristically decided to analyse the following defined symbols:
activate', sel'

They will be analysed ascendingly in the following order:
activate' < sel'

Generator Equations:
_gen_n__from':cons':nil':n__first'3(0) ⇔ n__from'(0')
_gen_n__from':cons':nil':n__first'3(+(x, 1)) ⇔ cons'(0', _gen_n__from':cons':nil':n__first'3(x))
_gen_s':0'4(0) ⇔ 0'
_gen_s':0'4(+(x, 1)) ⇔ s'(_gen_s':0'4(x))

The following defined symbols remain to be analysed:
activate', sel'

They will be analysed ascendingly in the following order:
activate' < sel'

Could not prove a rewrite lemma for the defined symbol activate'.

The following defined symbols remain to be analysed:
sel'

Could not prove a rewrite lemma for the defined symbol sel'.

The following conjecture could not be proven:

sel'(_gen_s':0'4(_n32), _gen_n__from':cons':nil':n__first'3(1)) →^? _*5

No more defined symbols left to analyse.